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SIGMA, 2012, Volume 8, 100, 53 pages (Mi sigma777)  

This article is cited in 13 scientific papers (total in 13 papers)

Geometry of Spectral Curves and All Order Dispersive Integrable System

Gaëtan Borota, Bertrand Eynardbc

a Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland
b Institut de Physique Théorique, CEA Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France
c Centre de Recherche Mathématiques de Montréal, Université de Montréal, P.O. Box 6128, Montréal (Québec) H3C 3J7, Canada

Abstract: We propose a definition for a Tau function and a spinor kernel (closely related to Baker–Akhiezer functions), where times parametrize slow (of order $1/N$) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of $1/N$, where the coefficients involve theta functions whose phase is linear in $N$ and therefore features generically fast oscillations when $N$ is large. The large $N$ limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in $1/N$ an isomonodromic problem given by a Lax pair, and the relation between “correlators”, the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

Keywords: topological recursion; Tau function; Sato formula; Hirota equations; Whitham equations


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ArXiv: 1110.4936
MSC: 14H70; 14H42; 30Fxx
Received: November 14, 2011; in final form December 11, 2012; Published online December 18, 2012

Citation: Gaëtan Borot, Bertrand Eynard, “Geometry of Spectral Curves and All Order Dispersive Integrable System”, SIGMA, 8 (2012), 100, 53 pp.

Citation in format AMSBIB
\by Ga\"etan~Borot, Bertrand~Eynard
\paper Geometry of Spectral Curves and All Order Dispersive Integrable System
\jour SIGMA
\yr 2012
\vol 8
\papernumber 100
\totalpages 53

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    This publication is cited in the following articles:
    1. Kalla C., Korotkin D., “Baker-Akhiezer Spinor Kernel and Tau-Functions on Moduli Spaces of Meromorphic Differentials”, Commun. Math. Phys., 331:3 (2014), 1191–1235  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Alvarez G., Martinez Alonso L., Medina E., “Partition Functions and the Continuum Limit in Penner Matrix Models”, J. Phys. A-Math. Theor., 47:31 (2014), 315205  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Bergere M., Borot G., Eynard B., “Rational Differential Systems, Loop Equations, and Application To the Qth Reductions of Kp”, Ann. Henri Poincare, 16:12 (2015), 2713–2782  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Borot, G.; Eynard, B., “All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials”, Quantum Topology, 6:1 (2015), 39-138  crossref  mathscinet  zmath  elib  scopus
    5. G. Borot, S. Shadrin, “Blobbed topological recursion: properties and applications”, Math. Proc. Camb. Philos. Soc., 162:1 (2017), 39–87  crossref  mathscinet  zmath  isi  scopus
    6. R. Belliard, B. Eynard, O. Marchal, “Integrable differential systems of topological type and reconstruction by the topological recursion”, Ann. Henri Poincare, 18:10 (2017), 3193–3248  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. M. Marino, S. Zakany, “Exact eigenfunctions and the open topological string”, J. Phys. A-Math. Theor., 50:32 (2017), 325401  crossref  mathscinet  zmath  isi  scopus
    8. V. Bouchard, B. Eynard, “Reconstructing WKB from topological recursion”, Journal de l'Ecole Polytechnique - Mathematiques, 4 (2017), 845-908  crossref  mathscinet  scopus
    9. R. Belliard, B. Eynard, O. Marchal, “Loop equations from differential systems on curves”, Ann. Henri Poincare, 19:1 (2018), 141–161  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. K. Iwaki, O. Marchal, A. Saenz, “Painlevé equations, topological type property and reconstruction by the topological recursion”, J. Geom. Phys., 124 (2018), 16–54  crossref  mathscinet  zmath  adsnasa  isi  scopus
    11. N. Do, P. Norbury, “Topological recursion on the Bessel curve”, Commun. Number Theory Phys., 12:1 (2018), 53–73  crossref  mathscinet  zmath  isi  scopus
    12. V. Bouchard, N. K. Chidambaram, T. Dauphinee, “Quantizing weierstrass”, Commun. Number Theory Phys., 12:2 (2018), 253–303  crossref  mathscinet  zmath  isi  scopus
    13. Marino M., Zakany S., “Wavefunctions, Integrability, and Open Strings”, J. High Energy Phys., 2019, no. 5, 014  crossref  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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